Geoscience Reference
In-Depth Information
Substituting(7.4)into(7.2)andgroupingterms
tQ
i
A
i
θ
i
+
1
,
j
+
1
+
B
i
θ
i
,
j
+
1
+
C
i
θ
i
−
1
,
j
+
1
=
θ
i
,
j
−
1
+
2
∆
(7.5)
where
tK
θ
i
+
1
tK
θ
i
A
i
=
−
2
∆
C
i
=
−
2
∆
B
i
=
1
−
A
i
−
C
i
∆
z
i
∆
zz
i
∆
z
i
∆
zz
i
−
1
Fromhereon,thesecondindexreferringtothenew
(
j
+
1
)
timestepwillbedropped
from the notation, with only the previous
(
j
−
1
)
time step indicated explicitly. To
usetheimplicit solutiontechnique,let
θ
i
=
D
i
+
E
i
θ
i
+
1
(7.6)
sothat
tQ
i
A
i
θ
i
+
1
+
B
i
θ
i
+
C
i
(
D
i
−
1
+
E
i
−
1
θ
i
)=
θ
i
,
j
−
1
+
2
∆
fromwhich
tQ
θ
i
−
=
−
A
i
θ
i
+
1
+
θ
i
,
j
−
1
+
2
∆
C
i
D
i
−
1
θ
(7.7)
i
B
i
+
C
i
E
i
−
1
hencetherecursionrelationis
A
i
E
i
=
(7.8)
A
i
−
1
+
C
i
(
1
−
E
i
−
1
)
C
i
D
i
−
1
−
θ
i
,
j
−
1
+
tQ
i
2
∆
D
i
=
A
i
−
1
+
C
i
(
1
−
E
i
−
1
)
This approach holds as well for the momentumconservationequation in a rotating
reference frame provided the Coriolis term is treated as a source (time centered in
theleapfrogscheme).Usingcomplexnotationforthehorizontalvelocity(relativeto
undisturbedgeostrophicflow) in a horizontallyhomogeneousfluid,themomentum
balanceis
u
t
=
τ
z
−
if
u
(7.9)
where
τ
z
is the verticalgradientof the kinematicReynoldsstress in the fluid. Note
that an additional source term (e.g., a constant or depth varying [“thermal wind”]
geostrophiccurrent)maybeaddedto theCoriolissourceterm.
7.2 Boundary Conditions
The starting point for the recursion relations (7.8) is provided by consideration of
theboundaryconditionsforthedifferenceformoftheconservationequationforthe
genericproperty
θ
.